Discrete Uniform Distribution: Mean, Variance, MGF
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Questions and Answers

What is the probability mass function (PMF) of a uniform distribution U(a,b)?

  • $P(X=x)=\frac{1}{2(b-a+1)}$
  • $P(X=x)=\frac{1}{b-a+1}$ (correct)
  • $P(X=x)=\frac{1}{b+a-1}$
  • $P(X=x)=\frac{1}{a+b+1}$

What is the variance of a uniform distribution U(a,b)?

  • $Var(X)=\frac{(b+a-1)^2}{12}$
  • $Var(X)=\frac{a+b+1}{12}$
  • $Var(X)=\frac{(b-a+1)^2}{12}$ (correct)
  • $Var(X)=\frac{(b-a)^2}{12}$

What is the moment generating function (MGF) of a uniform distribution U(a,b)?

  • $MX(t)=(a+b+1)(e^t-1)e^{at}-e^{(2b)t}$
  • $MX(t)=(b-a)(e^t-1)e^{at}-e^{bt}$
  • $MX(t)=(b+a-1)(e^t-1)e^{at}-e^t$
  • $MX(t)=(b-a+1)(e^t-1)e^{at}-e^{(b+1)t}$ (correct)

For a uniform distribution U(2,8), what is the mean?

<p>$3$ (A)</p> Signup and view all the answers

What is the variance of the uniform distribution U(4,9)?

<p>$Var(X)=\frac{6^2}{12}$ (D)</p> Signup and view all the answers

Calculate the moment generating function for the uniform distribution U(3,7). What is the result?

<p>$(7-3+1)(e^t-1)e^{3t}-e^{8t}$ (D)</p> Signup and view all the answers

In a uniform distribution U(0,5), what is the probability of X taking a value of 3?

<p>$\frac{1}{5}$ (A)</p> Signup and view all the answers

Find the variance for the uniform distribution U(6,10). What is the correct variance calculation?

<p>$Var(X)=\frac{5^2}{12}$ (C)</p> Signup and view all the answers

In a uniform distribution U(3,9), what is the probability of X being 7?

<p>$\frac{1}{6}$ (B)</p> Signup and view all the answers

Calculate the mean for the uniform distribution U(5,11). What is the correct mean value?

<p>$8$ (C)</p> Signup and view all the answers

What is the mean and variance of a Poisson distribution?

<p>Mean: ?=?, Variance: ?2=? (A), Mean: ?=?, Variance: ?2=? (C), Mean: ?=?, Variance: ?2=? (D)</p> Signup and view all the answers

What is the moment generating function (MGF) of a Binomial distribution?

<p>$MX(t)=e^{(0.7+0.21e^t+0.0153e^{2t})}$ (C)</p> Signup and view all the answers

In the Poisson distribution, what does the parameter ? represent?

<p>Average rate of occurrence of events (C)</p> Signup and view all the answers

What does the Probability Mass Function (PMF) represent in a Binomial distribution?

<p>$P(X=k)=(kn?)p^k(1-p)^{n-k}$ (C)</p> Signup and view all the answers

What is the formula for the moment generating function (MGF) of a Poisson distribution?

<p>$MX(t)=e^{\theta(e^t-1)}$ (B)</p> Signup and view all the answers

What does the binomial coefficient (kn?) represent in a Binomial distribution PMF?

<p>$\frac{n!}{k!(n-k)!}$ (B)</p> Signup and view all the answers

In the context of moment generating functions, what does E[etX] stand for?

<p>$MX(t)$ (B)</p> Signup and view all the answers

Study Notes

Discrete Probability Distributions

Uniform Distribution (Discrete)

  • Notation: U(a, b) where a and b are the minimum and maximum values of the distribution
  • Probability Mass Function (PMF): 1/(b-a+1) for x=a, a+1, ..., b, and 0 for other values of x
  • Mean (μ): (a+b)/2
  • Variance (σ²): (b-a+1)²/12
  • Moment Generating Function (MGF): (b-a+1)(e^t-1) / (e^(b+1)t - e^at)

Example: Uniform Distribution U(1, 6)

  • Mean: 3.5
  • Variance: 35/12
  • Moment Generating Function: (6(e^t-1)) / (e^7t - e^t)

Binomial Distribution

  • Notation: B(n, p) where n is the number of trials and p is the probability of success
  • Probability Mass Function (PMF): (k choose n) p^k (1-p)^(n-k) where k is the number of successes
  • Mean (μ): np
  • Variance (σ²): np(1-p)
  • Moment Generating Function (MGF): Σ(etk) (k choose n) p^k (1-p)^(n-k) from k=0 to n

Example: Binomial Distribution B(5, 0.3)

  • Mean: 1.5
  • Variance: 1.05
  • Moment Generating Function: (0.7 + 0.21e^t + 0.0153e^(2t))

Poisson Distribution

  • Notation: Poisson(λ)
  • Probability Mass Function (PMF): e^(-λ) (λ^k) / k! where k is the number of events
  • Mean (μ): λ
  • Variance (σ²): λ
  • Moment Generating Function (MGF): e^(λ(e^t-1))

Example: Poisson Distribution Poisson(2)

  • Mean: 2
  • Variance: 2
  • Moment Generating Function: e^(2(e^t-1))

Continuous Probability Distributions

Continuous Uniform Distribution

  • A continuous uniform distribution is a probability distribution where every value between a certain range has an equal likelihood of occurring.

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Description

Learn about the properties of the discrete uniform distribution including notation, probability mass function, mean, variance, and moment generating functions. Understand how to calculate the mean and variance of a uniform distribution with minimum (a) and maximum (b) values.

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